Funnel object into another
So here is a solution with the Mesh Deform modifier:


Create your object.
Create a cube that contains your object as a cage, subdivide it. In the Properties panel > Object > Display, enable Draw All Edges and Maximum Draw Type > Wire so that your cube is in wire mode.
Give your object a Mesh Deform modifier. Put it above the Subdivision Surface modifier. Choose the cube as the Object, and click Bind.
Select the cube. In the Properties panel > Data > Shape Keys, create one Shape Key (it will be called Basis by default), then a second one.
Keep this second Shape Key selected. Go in Edit mode. Sculpt your cube (with Proportional Editing activated) so that it deforms your object the way you want.
Come back in Object mode. Now you can play with the Value of each Shape Key and insert Key Frames so your object is deformed the way you want at the moment you want on the Timeline.